21.Electron Bandslampx.tugraz.at/~hadley/ss1/lectures18/jun12.pdf · c H c H ihka jka lka ... Draw...
Transcript of 21.Electron Bandslampx.tugraz.at/~hadley/ss1/lectures18/jun12.pdf · c H c H ihka jka lka ... Draw...
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Institute of Solid State PhysicsTechnische Universität Graz
21.Electron Bands
June 12, 2018
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Tight binding
Tight binding does not include electron-electron interactions
222
02 4A
MOAe A
Z eHm r r
1 2 3 unit_cell 1 2 3, ,
1 expkl m n
i lk a mk a nk a r la ma naN
unit_cell i i ii
r c r r
Atomic wave functionsThis is the tight-binding wave function.
1 2 3exppqs k kT i pk a qk a sk a
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Tight binding, one atomic orbital
For only one atomic orbital in the sum over valence orbitals
1 2 3nearest neighbors
exp( ( ))k a a a a a MO a a a MO mm
E c c H c H i hk a jk a lk a
mikk
mE t e
a MO ar H r
a MO a mt r H r
1 2 3nearest neighbors
exp( ( )) small terms
small terms
a a MO a m a MO mm
k a a a
c H c H i hk a jk a lk a
E c
On-site energy Overlap integral
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Tight binding, fcc
lmnik
lmn
E t e
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Density of states (fcc)
Calculate the energy for every allowed k in the Brillouin zone
http://lamp.tu-graz.ac.at/~hadley/ss1/bands/tbtable/tbtable.html
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Tight binding, fcc
Christian Gruber, 2008
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Tight binding, fcc
http://www.phys.ufl.edu/fermisurface/
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http://lamp.tu-graz.ac.at/~hadley/ss1/bands/tbtable/tbtable.html
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http://www.springermaterials.com/navigation/bookshelf.html#l_2_106048_
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Tight binding
Harrison, Electronic Structure, Freeman 1980
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Tight binding 1-d
Finite potential well
bTight binding fails when the electrons aren't mostly localized.
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Tight binding 1-d
k = 0
k = /a
k = 0
first band
second band
2N electron states per band
( ) ikna
n
x e x na
k = /a
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2 electron states/band
The N atomic orbitals in the tight binding wave function can hold 2Nelectrons
N atoms in the crystal
N translational symmetries
N allowed k states in the first Brillouin zone
these k's can be put in one-to-one correspondence with the eigenvalues of the translation operator
N solutions to the Schrödinger equation of Bloch form labeled by k
2N electron states in the first Brillouin zone
2N electrons per band
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2N states per Brillouin zone
A crystal L×L×L has unit cells.3
3
LNa
The first Brillouin zone contains k points.
3
3
3 3
2
2LaNa
L
There are N translational symmetries.
Each k state can hold 2 electrons (spin).
There are 2N per Brillouin zone.
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Thermodynamic properties of metals
From the band structure measurements, we obtain the electron density of states.
Thermodynamic properties can be calculated from the tabulated data for the density of states
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Metals, semiconductors, and insulators
From Ibach & Lueth
Insulators: band gap > 3 eV
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Student Projects
Plot the dispersion relation for a 1-D square wave potential Using the Kronig-Penney model, the plane wave method, and tight binding.
Draw the missing Fermi surfaces
Use the tight-binding model to calculate a dispersion relation.
Make a similar table to the table of tight binding solutionsfor the plane wave method
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Institute of Solid State PhysicsTechnische Universität Graz
Crystal Physics
Crystal physics explains what effects the symmetries of the crystal have on observable quantities.
An Introduction to Crystal Physics Ervin Hartmannhttp://ww1.iucr.org/comm/cteach/pamphlets/18/index.html
International Tables for Crystallographyhttp://it.iucr.org/
Kittel chapter 3: elastic strain
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Strain
A distortion of a material is described by the strain matrix
ˆ ˆ ˆ(1 )ˆ ˆ ˆ(1 )ˆ ˆ ˆ(1 )
xx xy xz
yx yy yz
zx zy zz
x x y z
y x y z
z x y z
x
y
z
x
y
z
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Stress
9 forces describe the stress
Xx, Xy, Xz, Yx, Yy, Yz, Zx, Zy, Zz
Xx is a force applied in the x-direction to the plane normal to x
Xy is a sheer force applied in the x-direction to the plane normal to y
yx z
x y z
yx z
x y z
yx z
x y z
XX XA A A
YY YA A A
ZZ ZA A A
stress tensor:
Stress is force/m2
Xx Xy
Xz
x
y
z
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Stress and Strain
The stress - strain relationship is described by a rank 4 stiffness tensor. The inverse of the stiffness tensor is the compliance tensor.
ij ijkl kls
ij ijkl klc
Einstein convention: sum over repeated indices.
xx xxxx xx xxxy xy xxxz xz xxyx yx xxyy yy
xxyz yz xxzx zx xxzy zy xxzz zz
s s s s s
s s s s
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Statistical Physics
Microcannonical Ensemble: Internal energy is expressed in terms of extrinsic quantities U(S, M, P, , N, V).
ij ij k K l ldU TdS d E dP H dM
ij K lij k l
U U U UdU dS d dP dMS P M
The normal modes must be solved for in the presence of electric and magnetic fields (Advanced Solid State Physics course).
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Internal energy in an electric field
In an electric field, if the dipole moment is changed, the change of the energy is,
U E P
Using Einstein notation
This is part of the total derivative of U
ij ij k K l ldU TdS d E dP H dM
k kdU E dP
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Statistical Physics
Microcannonical Ensemble: Internal energy is expressed in terms of extrinsic quantities U(S, M, P, , N, V).
Cannonical ensemble: At constant temperature, make a Legendre transformation to the Helmholtz free energy.
F = U - TS
F(V, T, N, M, P, )
Make a Legendre transformation to the Gibbs potential G(T, H, E, )
ij ij k K l lG U TS E P H M
ij K lij k l
ij ij k K l l
U U U UdU dS d dP dMS P M
dU TdS d E dP H dM
ij ijV
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Gibbs free energy
ijij
G
kk
G PE
ll
G MH
G ST
ij ij k k l ldG SdT d P dE M dH
ij k lij k l
G G G GdG dT d dE dHT E H
ij ij k K l lG U TS E P H M
ij ij k K l ldU TdS d E dP H dM
ij ij ij ij k k k k l l l ldG dU TdS SdT d d E dP P dE H dM M dH
total derivative:
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